A method of constructing solutions of semilinear dissipative equations
in bounded domains is proposed. It allows to calculate the higher-order long-time
asymptotics. The application of this approach is given for solving the first initialboundary
value problem for the damped Boussinesq equation

in a unit ball $B$. Homogeneous boundary conditions and small initial data are
examined. The existence of mild global-in-time solutions is established in the space
$C^0([0,\infty), H^s_0(B)), s < 3/2$, and the solutions are constructed in the form of the
expansion in the eigenfunctions of the Laplace operator in $B$. For $ -3/2 +\varepsilon \leq s <3/2$, where $\varepsilon
> 0$ is small, the uniqueness is proved. The second-order long-time
asymptotics is calculated which is essentially nonlinear and shows the nonlinear
mode multiplication.